Four congruent semicircles are inscribed in a square of side length $1$ so that their diameters are on the sides of the square, one endpoint of each diameter is at a vertex of the square, and adjacent semicircles are tangent to each other. A small circle centered at the center of the square is tangent to each of the four semicircles, as shown below. The diameter of the small circle can be written as $(\sqrt a+b)(\sqrt c+d)$, where $a$, $b$, $c$, and $d$ are integers. What is $a+b+c+d$?
$\textbf{(A) } 3 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 11$
Difficulty: 51/100 - A standard yet somewhat tricky tangent circle problem.
Core Concepts: tangent circles, special right triangles, Pythagorean theorem, radical simplification, quadratic
Challenges: Even after connecting the centers of the semicircles, it may not be immediately obvious to compute the radius of the semicircle. Determining the diameter of the inscribed circle after finding the radii of the semicircles may also not be apparent.
Techniques: Using the tangent circles and the fact that all semicircles have the same radius to form equations relating radii. Factoring the expression once an answer is reached.
Error-prone Steps: Errors with radical simplification, ineffective setups involving a lot of steps, taking the wrong solutions from the quadratics.
Ideal Time:
Experienced: ≤ 1-2 min
Intermediate: ≤ 4-5 min
Beginner: ≤ 10-15 min